Conformal Representation
Conformality and Uniformization By an isogonal representation of two areas on one another we mean a one-one, continuous, and continuously differentiable representation of the areas, which is such that two curves of the first area which intersect at an angle \( \alpha \) are transformed into two curves intersecting at the same angle \( \alpha . \) If the sense of rotation of a tangent is preserved, an isogonal transformation is called conformal. Disregarding as trivial the Euclidean magnification of the plane, we may say that the oldest known transformation of this kind is the stereographic projection of the sphere, which was devised by Ptolemy for the representation of the celestial sphere; it transforms the sphere conformally into a plane. A quite different conformal representation of the sphere on a plane area is given by Mercator's Projection; in this the spherical earth, cut along a meridian circle, is conformally represented on a plane strip. Uniformisation Theorem: The un...